I read this quote by J.K. Rowling from a 2004 interview:
[Snape] was a Death Eater, so clearly he is no Muggleborn, because Muggleborns are not allowed to be Death Eaters, except in rare circumstances.
[J.K. Rowling at the Edinburgh Book Festival 08.15.04]
Under what circumstances would a Muggleborn be allowed to become a Death Eater, and furthermore why would a Muggleborn consider joining up?
Voldemort was a half-blood, so was Snape.
Clearly, pure-bloodness was (probably intentionally, as the whole idea was meant by JKR to parallel Nazi laws) treated the same way as it was treated in Nazi Germany: as was the case with Erhard Milch, when a useful individual was personally approved by top brass, "Wer Jude ist, bestimme ich", to quote Hermann Göring. Heck, they made an exception for Fenrir Greyback despite the whole anti-werewolf thing.
As far as why would the muggleborn want to join? Again, going with JKR parallel, do I need to give you the list of Jews who served Nazis (or profess their deep and abiding love of Mahmoud Ahmadinejad)?
Or someone could simply be a person with no principles. Death Eaters seems to be the winning side - why NOT join if you manage to finagle your way in?
There are plenty of real-world examples of converts to a religion or ideology becoming even bigger fanatics than people who were born into it. I could see a Muggleborn wizard vying for favor with the death-eaters, being a vehement anti-Muggle blowhard, proclaiming his own "conversion", or describing his muggle blood as a weakness but still swearing loyalty to the "cause".
Why would the death eaters accept that wizard? He might be useful. He might be accepted for his contacts or usefulness, with the intention of killing him once those are outgrown or used up. Maybe his blood can be ignored or smoothed over, if not rewritten outright, if he is powerful or useful enough.
You can look in history for the sort of racist organizations that exist in reality, from which Rowling certainly borrowed ideas and inspiration.
One of the ways that Harry's mom, a muggle-born witch, defied Voldemort was by refusing to join the Death Eaters. Voldemort himself asked Lily and James to join him, so he wasn't really looking for blood purity at the time, just talent.
Well if they happened to have one of those pureblood names (such as Potter or Black which is extremely common with Muggles) then they could probably skim over having "dirty blood" by claiming they are illegitimate or a distant cousin or something, so the pureblood issue would be solved. I can also think of many reasons why a muggle-born would join an anti-muggle group, such as abuse, rape, discrimination, etc. Plus, for the eleven years before Harry Potter, Voldemort had practically won, so who wouldn't join up with him to save themselves? And, as seen within the series, halfbloods and mudbloods are usually more powerful than purebloods (cough, cough, inbreeding).
